We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension n with normal bundles isomorphic to${\mathcal{O}}_{{\mathbb{P}}^1}{(1)}^{\oplus p}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}^{\oplus (n-1-p)}$ for some nonnegative integer p. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n\le 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role.

我们研究不可弯曲的有理曲线，即具有同构的法丛的n维复流形中的非奇异有理曲线${\mathcal{哦}}_{{\mathbb{磷}}^1}{(1)}^{\oplus 磷}\oplus {\mathcal{哦}}_{{\mathbb{磷}}^1}^{\oplus (n-1-磷)}$对于一些非负整数p。众所周知的例子来自代数几何，作为无规则射影流形的一般最小有理曲线。在描述了不可弯曲有理曲线的变形空间上自然分布的微分几何性质与其各种极小有理切线的射影几何性质之间的关系后，我们集中讨论$磷=1$ 和 $n\le 5$，这是最简单的非平凡情况。在这种情况下，不可弯曲的有理曲线族基本上分为两类：Goursat 型或 Cartan 型。Goursat 类型的那些来自常微分方程，而 Cartan 类型的那些具有与接触几何相关的特殊特征。我们证明了任何非奇异立方 4 折上的线族是 Goursat 型的，而一般四次 5 折上的线族是 Cartan 型的，证明了这些最小有理切线变体的射影几何起关键作用。